Elements of Algebra

396 ELEMENTS PART II. this latter term to such as are composed of two or more of the smallest numbers of either kind; so that the simple numbers of the first kind will be 2, 3, 5, 11, 29; and the compound numbers of the same class will be 8, 12, 14, 18, 20, 27, 30, 32, 35, 40, 45, 48, 50, &c. The simple numbers of the second class will be 1, 7, 31; and all the rest of this class will be compound numbers; namely, 4, 6, 9, 10, 15, 16, 22, 24, 25, 28, 33, 36, 40, 42, 49. CHAP. XII. Of the Transformation of the Formula $ax^2 + cy^2$ into Squares, and higher Powers. 181. We have seen that it is frequently impossible to reduce numbers of the form $ax^2 + cy^2$ to squares; but whenever it is possible, we may transform this formula into another, in which $a=1$. For example, the formula $2p^2 - q^2$ may become a square; for, as it may be represented by $(2p+q)^2 - 2(p+q)^2$, we have only to make $2p+q=x$, and $p+q=y$, and we shall get the formula $x^2 - 2y^2$, in which $a=1$, and $c=2$. A similar transformation always takes place, whenever such formulæ can be made squares. Thus, when it is required to transform the formula $ax^2 + cy^2$ into a square, or into a higher power, (provided it be even) we may, without hesitation, suppose $a=1$, and consider the other cases as impossible. 182. Let, therefore, the formula $x^2 + cy^2$ be proposed, and let it be required to make it a square. As it is composed of the factors $(x+y\sqrt{-c}) \times (x-y\sqrt{-c})$, these factors must either be squares, or squares multiplied by the same number. For, if the product of two numbers, for example, $pq$, must be a square, we must have $p=r^2$, and $q=s^2$; that is to say, each factor is of itself a square; or $p=mr^2$, and $q=ms^2$; and therefore these factors are squares multiplied both by the same number. For which reason, let us make $x+y\sqrt{-c} = m(p+q\sqrt{-c})^2$; it will follow that $x-y\sqrt{-c} = m(p-q\sqrt{-c})^2$, and we shall have $x^2+cy^2 = m^2(p^2+cq^2)^2$, which is a square.